All stators output AC. DC regulators contain circuitry to convert the AC to DC. DC regulators require some sort of energy storage device to work properly. This can be either a battery or a capacitor. Trailtech.net sells both.
Blinkers, fans, horns and GPSs require DC. If you use a capacitor instead of a battery then they won't work when the motor is stopped.
Some of the aftermarket stators use two outputs so that there is less stress on the regulators. They suggest running one stator coil to a DC system so that you can have blinkers, etc. They suggest the other coil go to an AC system because that is more power efficient and requires fewer parts.
The simplest setup for you would be a stator with a single winding driving a DC system. This is what I have. The regulator is the common Tympanium unit sold by Baja Designs, and it uses a capacitor for storage. This is with a home-wound stator. If you buy a stator, I would use whatever regulator(s) the mfg recommends. Power ratings on stators don't tell you much unless you are comparing ratings from the same manufacturer.
For a multi-winding system, it would make the most sense to run a headlight off one winding with an AC regulator and everything else off the other winding with a DC regulator.
If your blinkers don't work with a flat battery replace the flasher unit with a better one. I have a $5 electronic (not thermal) flasher from Kragen that replaces the one from a DS kit, and it works fine below 10 volts.